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EEE Electrical Properties of Materials Lecture 2

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Bravais Lattice zThe unit cell of a general 3D lattice is described by 6 numbers z3 distances (a, b, c) z 3 angles (,, ) 27-Dec-132

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Bravais Lattice z There are 14 distinct 3D lattices which come under 7 Crystal Systems zThe BRAVAIS LATTICES (with shapes of unit cells as) : Cube (a = b = c, = = = 90 ) Square Prism (Tetragonal) (a = b c, = = = 90 ) Rectangular Prism (Orthorhombic) (a b c, = = = 90 ) 120 Rhombic Prism (Hexagonal) (a = b c, = = 90, = 120 ) Parallelepiped (Equilateral, Equiangular)(Trigonal) (a = b = c, = = 90 ) Parallelogram Prism (Monoclinic) (a b c, = = 90 ) Parallelepiped (general) (Triclinic) (a b c, ) 27-Dec-133

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Bravais Lattice zThe lattice centering are: zPrimitive (P): lattice points on the cell corners only. zBody (I): one additional lattice point at the center of the cell. zFace (F): one additional lattice point at the center of each of the faces of the cell. zBase (A, B or C): one additional lattice point at the center of each of one pair of the cell faces. 27-Dec-134

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Bravais Lattice 27-Dec-135

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Miller Indices 27-Dec-136 The planes passing through lattice points are called lattice planes. The orientation of planes or faces in a crystal can be described in terms of their intercepts on the three axes Miller introduced a system to designate a plane in a crystal. He introduced a set of three numbers to specify a plane in a crystal. This set of three numbers is known as Miller Indices of the concerned plane.

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Miller Indices 27-Dec-137 Miller indices is defined as the reciprocals of the intercepts made by the plane on the three axes.

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Miller Indices 27-Dec-138 Procedure for finding Miller Indices Step 1: Determine the intercepts of the plane along the axes X,Y and Z in terms of the lattice constants a,b and c. Step 2: Determine the reciprocals of these numbers.

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Miller Indices 27-Dec-139 Step 3: Find the least common denominator (lcd) and multiply each by this lcd. Step 4:The result is written in parenthesis. This is called the `Miller Indices of the plane in the form (h k l).

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Miller Indices 27-Dec-1310 Plane ABC has intercepts of 2 units along X-axis, 3 units along Y-axis and 2 units along Z-axis. PLANES IN A CRYSTAL ILLUSTRATION

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Miller Indices 27-Dec-1311 DETERMINATION OF MILLER INDICES Step 1:The intercepts are 2,3 and 2 on the three axes. Step 2:The reciprocals are 1/2, 1/3 and 1/2. Step 3:The least common denominator is 6. Multiplying each reciprocal by lcd, we get, 3,2 and 3. Step 4:Hence Miller indices for the plane ABC is (3 2 3) ILLUSTRATION

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Miller Indices 27-Dec-1312 Example In this plane, the intercept along X axis is 1 unit. The plane is parallel to Y and Z axes. So, the intercepts along Y and Z axes are. Now the intercepts are 1, and. The reciprocals of the intercepts are = 1/1, 1/ and 1/. Therefore the Miller indices for the above plane is (1 0 0).

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Miller Indices 27-Dec-1313 SOME IMPORTANT PLANES

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Miller Indices 27-Dec-1314 Problem # 1 A certain crystal has lattice parameters of 4.24, 10 and 3.66 Å on X, Y, Z axes respectively. Determine the Miller indices of a plane having intercepts of 2.12, 10 and 1.83 Å on the X, Y and Z axes. Lattice parameters are = 4.24, 10 and 3.66 Å The intercepts of the given plane = 2.12, 10 and 1.83 Å i.e. The intercepts are, 0.5, 1 and 0.5. Step 1:The Intercepts are 1/2, 1 and 1/2. Step 2:The reciprocals are 2, 1 and 2. Step 3:The least common denominator is 2. Step 4:Multiplying the lcd by each reciprocal we get, 4, 2 and 4. Step 5:By writing them in parenthesis we get (4 2 4) Therefore the Miller indices of the given plane is (4 2 4) or (2 1 2).

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Miller Indices 27-Dec-1315 Problem # 2 Calculate the miller indices for the plane with intercepts 2a, - 3b and 4c the along the crystallographic axes. The intercepts are 2, - 3 and 4 Step 1:The intercepts are 2, -3 and 4 along the 3 axes Step 2: The reciprocals are 1/2, -1/3, 1/4 Step 3: The least common denominator is 12. Multiplying each reciprocal by lcd, we get 6 -4 and 3 Step 4: Hence the Miller indices for the plane is

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Classical Theory of Electrical and Thermal Conduction 27-Dec-1316

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Electrical & Thermal Conduction zElectrical conduction: Motion of charges (conduction electrons) in a material under the influence of an applied electric field zThermal conduction: Conduction of thermal energy from higher to lower temperature regions in a material zThermal conduction involves carrying of energy by conduction electrons. zGood electrical conductors are also good thermal conductors! 27-Dec-1317

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Next Week! QUIZ! 27-Dec-1318

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